There are often articles in the news that state that a certain country or hacker has been able to decrypt/hack highly protected systems: Some examples are the Lockheed Martin RQ-170 Sentinel that was downed in Iran and later said to be partly reverse engineered by China or the Snowden files that might have been hacked by Russia and China. There are plenty of other examples.

My question is now the following: How could this have been done considering the fact that a brute-force method would certainly not work because of the encryption strength and leaving other non-technical aspects like spies etc. out of consideration.

Electromagnetic attack — attacks based on leaked electromagnetic radiation, which can directly provide plaintexts and other information. Such measurements can be used to infer cryptographic keys using techniques equivalent to those in power analysis or can be used in non-cryptographic attacks, e.g. TEMPEST (aka van Eck phreaking or radiation monitoring) attacks.

Acoustic cryptanalysis — attacks that exploit sound produced during a computation (rather like power analysis).

Differential fault analysis — in which secrets are discovered by introducing faults in a computation.

Data remanence — in which sensitive data are read after supposedly having been deleted.

Row hammer — in which off-limits memory can be changed by accessing adjacent memory.

Optical - in which secrets and sensitive data can be read by visual recording using a high resolution camera, or other devices that have such capabilities.

Orin Kerr & Bruce Schneier have a recent paper out titled Encryption Workarounds, where they group the techniques to indirectly attack theoretically-secure encryption. I think their break-down into abstract categories is helpful, as it's freer of the technical "how", and you can miss the forest for the trees. Think of all the possible ways to get what you want, and you don't want to break the encryption, you just want what's being protected by said encryption.

In general, I also think it's a massive mistake to "[leave out] non-technical aspects like spies", because if it's technically secure, that's all you have to work with.

Find the (Encryption) Key

It could be written down in a notebook, or on a post-it. Or on a keylogger planted on a suspect's computer.

Guess the Key

Usually one key will be used to encrypt another, more secure key. Humans are terrible at generating/remembering entropy, while computers are increasingly fast.

Compel the Key

Ranging from torture to inconvenience at the border. The distinction between the 4th (search & seizure) and 5th (self-incrimination) amendments (in the US) also poses an interesting case where the government can compel you to provide your biometrics to unlock a device (a warrant obviates the 4th amendment), but maybe not a passphrase that you've remembered (runs afoul of the 5th). The court may lock you up indefinitely for contempt, however, as in the case of Francis Rawls. 5th amendment be damned.

Exploit a Flaw

Implementations aren't perfect, exploit those bugs. Ella's answer summarizes many of those side-channel attacks, but some are more akin to "finding the key" if memory is read.

Access Plaintext when in Use

People don't directly read encrypted data, they enter their password to unlock it. This technique was used on the Silk Road: stage a fight near the suspect that distracts him, then while he looks away, steal his unlocked laptop!

Locate a Plaintext Copy

The ultimate bypass: maybe there's an unencrypted copy of the data you want somewhere else entirely. The difficulty here is it might be different from the encrypted version and you won't necessarily know.